Topic 8 Appendix B: Derivation of the Equation used to Calculate Keq

Russell Larsen; Mouna Maalouf

Topic 8 Appendix B: Derivation of the Equation used to Calculate Keq

If the total hemoglobin subunit concentration is constant (it is either bound to imidazole or not), it follows that [Hb⁺]total = [Hb•Fe⁺] + [Hb•Fe – Im⁺], or [fraction not bound] + [fraction bound] = 1

$\displaystyle \frac{{{[Hb.Fe^{+}]}}}{[Hb^{+}]_{total}}$ $\displaystyle + \frac{{{[Hb.Fe-Im^{+}]}}}{[Hb^{+}]_{total}}$ $\displaystyle = {X_{Hb.Fe}} + {X_{Hb.Fe-Im^{+}} = 1$

where XHb•Fe⁺ is the fraction not bound and X Hb•Fe – Im⁺ is the fraction bound to imidazole. A₀ will be the absorbance of hemoglobin subunits not bound to imidazole, and A₁₀₀ will be the absorbance of hemoglobin subunits 100% bound to imidazole.
The observed absorbance, A, is equal to,

A = A₀X_Hb•Fe⁺+ A₁₀₀ X _{Hb•Fe – Im⁺}.

This expression can be solved for either X_Hb•Fe⁺ or for X _{Hb•Fe –Im⁺} as follows:

Case 1

Case 2

$\displaystyle \displaystyle {A} = {A}_{0}{X_{Hb.Fe^{+}}} + {A}_{100}{(1-X_{Hb.Fe^{+}})$

$\displaystyle \displaystyle {A} - {A}_{100} = ({A}_{0}-{A}_{100}){X_{Hb.Fe^{+}}}$

$\displaystyle{X_{Hb.Fe^{+}} = \displaystyle \frac{{{{A}-A_{100}}}}{{A}_{0}{-}{A}_{100}}}$

$\displaystyle \displaystyle {A} = {A}_{0}({1 - {X_{Hb.Fe-Im^{+}}}) + {A}_{100}{X_{Hb.Fe-Im^{+}}$

$\displaystyle \displaystyle {A} - {A}_{0} = ({-}{A}_{0} + {A}_{100}){X_{Hb.Fe-Im^{+}}}$

$\displaystyle{X_{Hb.Fe-Im^{+}} = \displaystyle \frac{{{{A}_{0}-A}}}{{A}_{0}{-}{A}_{100}}}$

$\displaystyle \displaystyle {A} = {A}_{0}({1 - {X_{Hb.Fe-Im^{+}}}) + {A}_{100}{X_{Hb.Fe-Im^{+}}$

$\displaystyle \displaystyle {A} - {A}_{0} = ({-}{A}_{0} + {A}_{100}){X_{Hb.Fe-Im^{+}}}$

$\displaystyle{X_{Hb.Fe-Im^{+}} = \displaystyle \frac{{{{A}_{0}-A}}}{{A}_{0}{-}{A}_{100}}}$

The equilibrium constant, K_eq, is equal to:

$\displaystyle Keq=\frac{{{1}}}{{[Im]}}*\frac{{{[Hb•Fe-Im^{+}]}}}{{[Hb•Fe^{+}]}}$ $\displaystyle =\frac{{{1}}}{{[Im]}}$ * $\displaystyle \frac{{{[{X}_{Hb.Fe-Im^{+}}]}}}{{[{X}_{Hb.Fe^{+}}]}}$ $\displaystyle =$

$\displaystyle \frac{{{1}}}{{[Im]}}$ * $\displaystyle \frac{{{({A}_{0}-A)}}}{{(A}_{0}{-}{A}_{100})}}$ $\displaystyle \frac{{{({A}_{0}-{A}_{100})}}}{{(A}{-}{A}_{0})}}$ $\displaystyle =\frac{{{1}}}{{[Im]}}$ * $\displaystyle \frac{{{({A}_{0}-A)}}}{{(A}{-}{A}_{100})}}$ .

Finally, the equilibrium constant can be written as:

$\displaystyle Keq=\frac{{{1}}}{{[Im]}}$ * $\displaystyle \frac{{{({A}-A_{0})}}}{{(A}_{100}{-}{A})}}$ .

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